The modular curve $X_{193h}$

Curve name $X_{193h}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $ \left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 10 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $6$ $X_{8}$
$4$ $12$ $X_{25}$
$8$ $48$ $X_{193}$
Meaning/Special name
Chosen covering $X_{193}$
Curves that $X_{193h}$ minimally covers
Curves that minimally cover $X_{193h}$
Curves that minimally cover $X_{193h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -28991029248t^{30} + 159450660864t^{28} - 338832654336t^{26} + 346080411648t^{24} - 196708663296t^{22} + 125306929152t^{20} - 102339182592t^{18} + 39537082368t^{16} - 6396198912t^{14} + 489480192t^{12} - 48024576t^{10} + 5280768t^{8} - 323136t^{6} + 9504t^{4} - 108t^{2}\] \[B(t) = 1899956092796928t^{45} - 15674637765574656t^{43} + 54861232179511296t^{41} - 105863178545528832t^{39} + 118606243433545728t^{37} - 62247682575433728t^{35} - 24458223843016704t^{33} + 69236144121839616t^{31} - 52965392812867584t^{29} + 21023932022784000t^{27} - 5553079449550848t^{25} + 1388269862387712t^{23} - 328498937856000t^{21} + 51724016418816t^{19} - 4225838874624t^{17} + 93300719616t^{15} + 14841004032t^{13} - 1767370752t^{11} + 98592768t^{9} - 3193344t^{7} + 57024t^{5} - 432t^{3}\]
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 30200268x - 37464430192$, with conductor $141120$
Generic density of odd order reductions $139/1344$

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